# Conformally related Riemannian metrics with non-generic holonomy

## TL;DR

This paper classifies conformal classes on compact manifolds that contain two non-homothetic metrics with non-generic holonomy, revealing specific geometric structures such as ambikähler, Calabi Ansatz, or reducible holonomy configurations.

## Contribution

It provides a comprehensive classification of conformal classes with two non-homothetic non-generic holonomy metrics, identifying their geometric structures and conditions after finite coverings.

## Key findings

- If n=4, the manifold is ambikähler.
- For even n≥6, the manifold arises from the Calabi Ansatz on a polarized Hodge manifold.
- Both metrics may have reducible holonomy and the manifold decomposes as a product with specific metric forms.

## Abstract

We show that if a compact connected $n$-dimensional manifold $M$ has a conformal class containing two non-homothetic metrics $g$ and $\tilde g=e^{2\varphi}g$ with non-generic holonomy, then after passing to a finite covering, either $n=4$ and $(M,g,\tilde g)$ is an ambik\"ahler manifold, or $n\ge 6$ is even and $(M,g,\tilde g)$ is obtained by the Calabi Ansatz from a polarized Hodge manifold of dimension $n-2$, or both $g$ and $\tilde g$ have reducible holonomy, $M$ is locally diffeomorphic to a product $M_1\times M_2\times M_3$, the metrics $g$ and $\tilde g$ can be written as $g=g_1+g_2+e^{-2\varphi}g_3$ and $\tilde g=e^{2\varphi}(g_1+g_2)+g_3$ for some Riemannian metrics $g_i$ on $M_i$, and $\varphi$ is the pull-back of a non-constant function on $M_2$.